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Gauss) in a geometrical shape which appeals particularly to
those who, like Poincar6? prefer the intuitive approach. This of
course was not all that he did in the higher arithmetic, but
limitations of space forbid further details.
Work of this calibre did not pass unappreciated. At the
unusually early age of thirty-two (in 1887) Poinear6 was elected
to the Academy. His proposer said some pretty strong things, but
most mathematicians will subscribe to their truth: t[Poincare'*s]
work is above ordinary praise and reminds us inevitably of
what Jacobi wrote of Abel - that he had settled questions
which, before him, were unimagined. It must indeed be recog-
nized that we are witnessing a revolution in Mathematics com-
parable m every way to that which manifested itself, fra-lf a
century ago, by the accession of elliptic functions.'
To leave Poineare's work in pure mathematics here is like
rising from a banquet table after having just sat down, but we
must turn to another side of his universality.
Since the time of Newton and his immediate successors astro-
nomy has generously supplied mathematicians with more pro-
blems than they can solve. Until the late nineteenth century the
weapons used by mathematicians in then attack on astronomy
were practically all immediate improvements of those invented
by Newton himself, Euler, Lagrange, and Laplace. But all
through the nineteenth century, particularly since Cauchy's
development of the theory of functions of a complex variable
and the investigations of himself and others on the convergence
of infinite series, a huge arsenal of untried weapons had been
accumulating from the labours of pure mathematicians. To
Poincare, to whom analysis came as naturally as thinking, this
vast pile of unused mathematics seemed the most natural thing
in the world to use in a new offensive on the outstanding pro-
blems of celestial mechanics and planetary evolution. He picked
and chose what he liked out of the heap,, improved it, invented
new weapons of his own, and assaulted theoretical astronomy
in a grand fashion it had not been assaulted in for a century. He
modernized the attack; indeed his campaign was so extremely
modern to the majority of experts in celestial mechanics that
even to-day, forty years or more after Poincare opened his